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Risk, Returns, and Risk Aversion

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Risk, Returns, and Risk Aversion Return and Risk Measures Real versus Nominal Rates EAR versus APR Holding Period Returns Excess Return and Risk Premium – PowerPoint PPT presentation

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Title: Risk, Returns, and Risk Aversion


1
Risk, Returns, and Risk Aversion
  • Return and Risk Measures
  • Real versus Nominal Rates
  • EAR versus APR
  • Holding Period Returns
  • Excess Return and Risk Premium
  • Variance
  • Sharpe Ratio
  • Risk Aversion and Capital Allocations
  • Risk Aversion and Utility Function
  • Capital allocation line
  • Optimal Allocations

2
Road map in this and the next lecture
  • Risk and return
  • Optimal allocation given risk and return tradeoff
  • Two-asset allocation
  • Efficient frontier
  • Multiple asset allocation
  • Capital asset pricing models (CAPM)
  • Arbitrage pricing theory (APT)
  • Fama-French three-factor model

3
Nominal and Real Rates
  • Nominal rate
  • Real rate

4
Example 5.2 Annualized Rates of Return
5
Formula for EARs and APRs
See page 128-129
6
Rates of Return Single Period

HPR Holding Period Return P0 Beginning
price P1 Ending price D1 Dividend during
period one
7
Rates of Return Single Period Example
  • Ending Price 48
  • Beginning Price 40
  • Dividend 2
  • HPR (48 - 40 2 )/ (40)

8
Excess Return
  • Risk free rate
  • Excess return
  • Also known as risk premium

9
Scenario or Subjective Returns Example
State Prob. of State r in State .1
-.05
.10 2 .2 .05 3 .4 .15 4 .2 .25 5 .1 .35
E(r)
10
Variance or Dispersion of Returns
Standard deviation variance1/2
Using Our Example
Var
11
Mean and Variance of Historical Returns
Arithmetic average or rates of return
12
Geometric Average Returns
TV Terminal Value of the Investment
g geometric average rate of return
13
Sharpe Ratio
Sharpe Ratio for Portfolios
Measure of risk-return tradeoff Other concepts
Skewness and Kurtosis page 142-143 Check out
the statistics from page 147-151
14
Page 187
15
Figure 5.4 The Normal Distribution
16
Figure 5.4 The Normal Distribution
17
Normality and Risk Measures
  • What if excess returns are not normally
    distributed?
  • Standard deviation is no longer a complete
    measure of risk
  • Sharpe ratio is not a complete measure of
    portfolio performance
  • Need to consider skew and kurtosis

18
Skew and Kurtosis
  • Skew
  • Kurtosis
  • Equation 5.19
  • Equation 5.20

19
Figure 5.5A Normal and Skewed Distributions
20
Figure 5.5B Normal and Fat-Tailed Distributions
(mean .1, SD .2)
21
Value at Risk (VaR)
  • A measure of loss most frequently associated with
    extreme negative returns
  • VaR is the quantile of a distribution below which
    lies q of the possible values of that
    distribution
  • The 5 VaR , commonly estimated in practice, is
    the return at the 5th percentile when returns are
    sorted from high to low.

22
Expected Shortfall (ES)
  • Also called conditional tail expectation (CTE)
  • More conservative measure of downside risk than
    VaR
  • VaR takes the highest return from the worst cases
  • ES takes an average return of the worst cases

23
Lower Partial Standard Deviation (LPSD)and the
Sortino Ratio
  • Issues
  • Need to consider negative deviations separately
  • Need to consider deviations of returns from the
    risk-free rate.
  • LPSD similar to usual standard deviation, but
    uses only negative deviations from rf
  • Sortino Ratio replaces Sharpe Ratio

24
Historic Returns on Risky Portfolios
  • Returns appear normally distributed
  • Returns are lower over the most recent half of
    the period (1986-2009)
  • SD for small stocks became smaller SD for
    long-term bonds got bigger

25
Historic Returns on Risky Portfolios
  • Better diversified portfolios have higher Sharpe
    Ratios
  • Negative skew

26
Figure 5.10 Annually Compounded, 25-Year HPRs
27
Risk Aversion
  • Risk Aversion
  • Risk Love
  • Risk Neutral
  • Utility function

28
Utility Function
Utility Function U E ( r ) 1/2 A s2 Where U
utility E ( r ) expected return on the asset
or portfolio A coefficient of risk aversion s2
variance of returns
29
Computing Utility Scores
If A2, then See page 168
30
Figure 6.2 The Indifference Curve
Page 166, Table 6.3
31
Allocating Capital Risky Risk Free Assets
  • Its possible to split investment funds between
    safe and risky assets.
  • Risk free asset proxy T-bills
  • Risky asset stock (or a portfolio)

32
Example Using Chapter 6.4 Numbers
The total market value of an initial portfolio is
300,000, of which 90,000 is invested in the
Ready Asset money market fund, a risk-free asset.
The remaining 210,000 is invested in risky
securities 113,400 in equity and 96,600 in
long-term bonds. Find the distribution of this
portfolio.
33
Expected Returns for Combinations
34
Combinations Without Leverage
?
?
?
35
Capital Allocation Line with Leverage
  • Borrow at the Risk-Free Rate and invest in stock.
  • Using 50 Leverage,
  • rc (-.5) (.07) (1.5) (.15) .19
  • ?c (1.5) (.22) .33

36
(No Transcript)
37
Table 6.5 Utility Levels
38
Optimal Portfolio
  • Maximize the mean-variance utility function
  • UE(R)-1/2As2
  • Based on the expressions for expected return and
    s, we have the expression for optimal allocation
    y
  • Example 6.4 (page 175)

39
Figure 6.6 Utility as a Function of Allocation to
the Risky Asset, y
40
Figure 6.7 Indifference Curves for U .05 and U
.09 with A 2 and A 4
41
Optimal Complete Portfolio on Indifference Curves
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